# Big math/science results within easy reach

#### jsilvela

##### Senior Member
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There are some impressive results in math / comp. sci. that don't require much knowledge to get to.
I was thinking of sharing them in a recent thread on AI, but that was not really the place.

I'm thinking of two results that are somewhat related and can be big eye openers:

1. The halting problem.

Turing proved in 1936 that it was not possible to create an algorithm (program) that would determine, given another program as input, whether that program would finish (i.e. not loop infinitely).
The huge take-away: some things are fundamentally in-computable.
Note that this was done before the advent of computers.

2. The Cantor diagonal method.

This is explainable to someone with high-school math.
Cantor proved that the real numbers were not "countable".
A set is defined to be countable if it can be put in "into" correspondence with the natural numbers.
A countable set could be infinite (countably infinite), if it could be placed in "1-to-1" correspondence with the natural numbers.

Examples of countably infinite sets: the even numbers; the odd numbers; fractions i.e. rational numbers.

Cantor used a surprisingly elementary method to prove that there was no possible "1-to-1" mapping between the real numbers and the natural numbers.

The huge take-away: There are different sizes of infinity.

Cantor's method has been used as the basis for things like the halting theorem (above) or Gödel's incompleteness theorem.

What are other examples of cool science or math within unexpectedly easy reach?

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#### BDWoody

##### Chief Cat Herder
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What are other examples of cool science or math within unexpectedly easy reach?

Not sure if this qualifies, but thought it was very cool when I first saw it.

OP

#### jsilvela

##### Senior Member
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Not sure if this qualifies, but thought it was very cool when I first saw it.

That's awesome! I hope their proof is accepted.

#### BDWoody

##### Chief Cat Herder
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I also just thought of when my father boggled my mind with this one:

1=.9bar

Edit for clarity.
.9bar = 0.999999999... I couldn't find a way to actually have a superscript, and ellipses I thought might be less clear, so, it is meant to represent 9s going on forever.

Nothing to do with pressure. Sorry 'bout that.
End edit.

The proof was simple enough for my young mind to grasp, but profound enough to leave quite the impression.

Last edited:
OP

#### jsilvela

##### Senior Member
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Euclid's proof of the infinitude of primes also has few pre-requisites.

That's another, like .9bar, that could impress an eager kid. Or an eager adult.

#### sergeauckland

##### Major Contributor
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I also just thought of when my father boggled my mind with this one:

1=.9bar

The proof was simple enough for my young mind to grasp, but profound enough to leave quite the impression.
Not come across that one. Could you elucidate please?

S.

OP

#### jsilvela

##### Senior Member
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Not come across that one. Could you elucidate please?

S.
Just going to comment on the typography, because in Europe we don't tend to elide the zero in numbers like 0.9, as is done in the US.
And, for periodic decimals, if we had MathJAX or KaTeX or such, we could write the "bar" on top.

.9bar = 0.99999... periodic.

Hope that helps, will let BDWoody hint at the proof

#### sergeauckland

##### Major Contributor
Forum Donor
Just going to comment on the typography, because in Europe we don't tend to elide the zero in numbers like 0.9, as is done in the US.
And, for periodic decimals, if we had MathJAX or KaTeX or such, we could write the "bar" on top.

.9bar = 0.99999... periodic.

Hope that helps, will let BDWoody hint at the proof
Thanks. And there was me trying to see how air pressure came into it...

S.

#### antcollinet

##### Master Contributor
Just going to comment on the typography, because in Europe we don't tend to elide the zero in numbers like 0.9, as is done in the US.
And, for periodic decimals, if we had MathJAX or KaTeX or such, we could write the "bar" on top.

.9bar = 0.99999... periodic.

Hope that helps, will let BDWoody hint at the proof
Never seen it, but is the proof that:

1-.9bar = .0bar ?

#### blueone

##### Major Contributor
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Thanks. And there was me trying to see how air pressure came into it...

S.
Thank was my first thought too, and then it came to me that air pressure wasn't involved.

#### mhardy6647

##### Master Contributor
Thanks. And there was me trying to see how air pressure came into it...

S.
9-bar is like x-bar (mean) -- I get it now.

#### BDWoody

##### Chief Cat Herder
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Not come across that one. Could you elucidate please?

Hope that helps, will let BDWoody hint at the proof

I'm quite sure now that my lack of clarity has been addressed, that no hints are needed.

OP

#### jsilvela

##### Senior Member
Forum Donor
I'm quite sure now that my lack of clarity has been addressed, that no hints are needed.

Never seen it, but is the proof that:

1-.9bar = .0bar ?

Well, I'd say that .0bar is very clearly exactly = 0
So, 0.9bar + 0.0bar = 0.9bar

But the intuition is right: somehow, we need a 1 there to add to the 0.9999... to push it into being exactly 1.
It can't be 0.1. But it can't be 0.01 either ... 0.001 is also too big ...
It just has to be 0, but this path seems a bit unclear.

IMO there are clearer proofs.

Curious: how did your father explain it to you?

#### BDWoody

##### Chief Cat Herder
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Curious: how did your father explain it to you?

let x=.9bar

10x=9.9bar
10x-9=.9bar

10x-9=x
9x=9
x=1

#### antcollinet

##### Master Contributor

Well, I'd say that .0bar is very clearly exactly = 0
So, 0.9bar + 0.0bar = 0.9bar
Let me explain it better.

1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.9999 = 0.0001

1 - 0.9bar = 0.0bar (A never ending sequence of nines results in a never ending sequence of 0's)

So since 0.0bar clearly equals 0 then 1 must = 0.9bar.

EDIT: @BDWoody 's works also

#### BDWoody

##### Chief Cat Herder
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EDIT: @BDWoody 's works also

My father was always good at teaching me about limits...

#### Ron Texas

##### Master Contributor
https://gizmodo.com/new-13-sided-shape-the-hat-tiles-aperiodic-monotile-1850268575

The pattern appears to not repeat.

#### Doodski

##### Grand Contributor
Forum Donor
It reminds me of a friends home brew beta glasses or whatever they where called. He's a electronic technologist and took a pair of sunglasses and put diodes inside pointing at the eyes. Then he had circuitry to adjust the flashing rate, duration and brightness etc. They made one feel veryyy strange. Lol. Your picture makes me feel strange when looking deep into it.

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